Representation and Approximation of Positivity Preservers

We consider a closed set S?? ⁿ and a linear operator

$\Phi \colon \mathbb{R}X_1,\ldots,X_n]\rightarrow \mathbb{R}X_1,\ldots,X_n]$

that preserves nonnegative polynomials, in the following sense: if f≥0 on S, then Φ(f)≥0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which concerns linear functionals on ?X ₁,…,X _n ]. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.